Method for determining rock physics relationships using computer tomograpic images thereof

ABSTRACT

A method for estimating a relationship between physical properties of a porous material from a sample thereof includes making a three dimensional tomographic image of the sample of the material. The image is segmented into pixels each representing pore space or rock grains. The image is divided into sub-volumes. A porosity is estimated for each sub-volume. At least one petrophysical parameter is modeled from the image of each sub-volume. A relationship between the porosity and the at least one modeled petrophysical parameter is determined by, e.g., a best-fit statistical method. The relationship and the modeled petrophysical parameter for each sub-volume are stored in a computer or displayed.

CROSS-REFERENCE TO RELATED APPLICATIONS

Not applicable.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to the field of estimating materialproperties of porous media. More specifically, the invention relates tomethods for estimating such properties using computer tomographic (CT)images of porous media such as subsurface rock formation.

2. Background Art

Estimating material properties such as effective elastic and shearmoduli, electrical resistivity and fluid transport properties of porousmedia, for example, mobility of hydrocarbon in subsurface rockformations, has substantial economic significance. Methods known in theart for identifying the existence of subsurface hydrocarbon reservoirs,including seismic surveying and well log analysis, need to besupplemented with reliable methods for estimating how fluids disposed inthe pore spaces of reservoir rock formations will flow over time inorder to characterize the economic value of such reservoir rockformations.

One method known in the art for estimating fluid transport properties isdescribed in U.S. Pat. No. 6,516,080 issued to Nur. The method describedin the Nur patent includes preparing a “thin section” from a specimen ofrock formation. The preparation typically includes filling the porespaces with a dyed epoxy resin. A color micrograph of the section isdigitized and converted to an n-ary index image, for example a binaryindex image. Statistical functions are derived from the two-dimensionalimage and such functions are used to generate three-dimensionalrepresentations of the rock formation. Boundaries can be unconditionalor conditioned to the two-dimensional n-ary index image. Desiredphysical property values are estimated by performing numericalsimulations on the three-dimensional representations. For example,permeability is estimated by using a Lattice-Boltzmann flow simulation.Typically, multiple, equiprobable three-dimensional representations aregenerated for each n-ary index image, and the multiple estimatedphysical property values are averaged to provide a result.

In performing the method described in the Nur patent, it is necessary toobtain samples of the rock formation and to prepare, as explained above,a section to digitize as a color image. Economic considerations make itdesirable to obtain input to fluid transport analysis more quickly thancan be obtained using prepared sections. Recently, devices forgenerating CT images of samples such as drill cuttings have becomeavailable. Such CT image generating devices (CT scanners) typicallyproduce three-dimensional gray scale images of the samples analyzed inthe scanner. Such gray scale images can be used, for example,essentially contemporaneously as drill cuttings are generated during thedrilling of a wellbore through subsurface rock formations.

Using images of samples of rock formations it is possible to obtainestimates of petrophysical parameters of the imaged rock sample, forexample, porosity, permeability, shear and bulk moduli, and formationresistivity factor. The foregoing parameters are typically distributedwithin ranges in each rock formation, and there may be determinablerelationships between such parameters such that determining oneparameter value can enable determining one or more of the otherparameters. One way to establish such relationship is to determine oneor more rock physics transforms. A rock physics transform is amathematical formula or algorithm that relates one property of a rockformation to another. Such transforms can be based on an idealizedmathematical model of rock, such as the differential effective mediumthat models rock as a solid with ideal-shape inclusions or theHertz-Mindlin model that models rock as a composite made of perfectelastic spheres. Such transforms can also be based on a sufficientnumber of experimental data (e.g., well log measurements or laboratorymeasurements) using a statistically fit expression that approximatessuch data. An example of the latter is the Raymer transform betweenporosity φ and the compressional wave (P-wave) velocity of the rock(V_(p)). The transform is the expression V_(p)=(1−φ)²V_(ps)+φV_(pf),where V_(ps) is the P-wave velocity in the mineral (matrix or solid)phase of the rock (e.g., quartz) and V_(pf) is the P-wave velocity inthe pore fluid (e.g., water). The elastic (compressional) wave velocityis directly related to the bulk K and shear G moduli by the expressionV_(p)=√{square root over ((K+4G/3)ρ)}, where ρ is the bulk density ofthe rock. The foregoing moduli can be obtained by laboratorymeasurement, and can also be obtained by calculations made from an imageof a rock sample. Another example is the relationship between theabsolute permeability k and the porosity φ of a rock formation calledthe Kozeny-Carman relation, represented by the expressionk=d²φ³/[72τ²(1−φ)²], where d is the mean rock grain size and τ is thepore tortuosity (typically represented by a number between 1 and 5). Yetanother example is Humble's relationship between the electricalresistivity formation factor F and the porosity φ, represented by theexpression F=a/φ^(m), where a and m are constants that are determinedexperimentally. As in the P-wave velocity example, the parameters thatenter these two equations, one for permeability and the other for theformation factor, can be obtained by laboratory measurement and also bycalculations based on an image of a rock sample. Instead of using thepermeability (k) and formation factor (F) equation examples above, onemay conduct a large number of laboratory tests on samples that representthe formation under examination. Alternatively, such data can beobtained by digital calculations on a digitally imaged rock sample.

Obtaining and calibrating rock physics transforms using physical samplesand using measurements made on actual rock samples requires extensivelaboratory and/or well measurements. There exists a need to use imagessuch as the foregoing CT scan images to estimate relationships betweenformation parameters without the need for extensive laboratory or wellmeasurements.

SUMMARY OF THE INVENTION

A method according to one aspect of the invention for estimating arelationship between physical properties of a porous material from asample thereof includes making a three dimensional tomographic image ofthe sample of the material. The image is segmented into pixels eachrepresenting pore space or rock grains. The image is divided intosub-volumes. A porosity is estimated for each sub-volume from the imagethereof. At least one petrophysical parameter is modeled from the imageof each sub-volume. A relationship between the porosity and the at leastone modeled petrophysical parameter is determined. The relationship andthe modeled petrophysical parameter for each sub-volume are stored in acomputer or displayed.

Other aspects and advantages of the invention will be apparent from thefollowing description and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example of obtaining cuttings during drilling of awellbore and analysis thereof during the drilling.

FIG. 2 shows a flow chart of an example process for CT imagesegmentation.

FIG. 3 shows a flow chart of an example analysis procedure according tothe invention.

DETAILED DESCRIPTION

The description of this invention is made with reference to using drillcuttings obtained during the drilling of a wellbore through subsurfaceformations. It should be clearly understood that drill cuttings is onlyone example of samples of rock formation that may be used with thepresent invention. Any other source of a rock formation sample, e.g.,whole cores, sidewall cores, outcrop quarrying, etc. may providesuitable samples for analysis using methods according to the invention.Consequently, the invention is not limited in scope to analysis of drillcuttings.

An example of drilling a wellbore to obtain samples of rock formationsfor evaluation by examples of a method according to the invention willbe explained with reference to FIG. 1. A drilling unit or “rig” 10 isdisposed at the Earth's surface. The rig 10 includes lifting equipment(not shown separately) for raising and lowering one of several types ofdevice used to rotate a drill string 14. The device, shown at 18 in thepresent example may be a top drive, although the use of a tope drive isnot a limit on the scope of the invention. The drill string 14 isassembled by threadedly coupling segments of drill pipe end to end. Adrill bit 16 is disposed at the lower end of the drill string 14 andcuts through subsurface rock formations 11 to form a wellbore 12. Duringthe drilling of the wellbore 12, the rig 10 is operated to cause some ofthe axial load (weight) of the drill string 14 to be applied to thedrill bit 16. The top drive 18 rotates the drill string 14 and the drillbit 16 at the lower end thereof. The combination of axial load androtation causes the drill bit 16 to cut through the formations 11.

The rig 10 includes a tank or pit 22 having drilling fluid (“mud”) 20stored therein. A pump 24 lifts the mud 20 and discharges it throughsuitable flow lines 26 so that the mud 20 passes through an internalpassage in the drill string 14, whereupon it is discharged throughsuitable orifices or courses in the drill bit 16. The discharged mud 20cools and lubricates the drill bit 16 and lifts the cuttings generatedby the bit 16 to the Earth's surface. The cuttings and mud thus liftedenter separation and cleaning devices, shown generally at 28 andincluding, for example, devices known as “degassers” and “shale shakers”to remove the cuttings and contamination from the mud 20. The mud aftersuch cleaning is returned to the pit 22 for subsequent use in drillingthe wellbore 12.

In the present example, the cuttings removed from the separation andcleaning device 28 may be transported to a computer tomographic (“CT”)scanner 30, which may use x-rays for analysis of internal structure ofthe cuttings, for generation of three dimensional (3D) images of thecuttings. The images so generated may be in numerical form and theircontent will be further explained below. After CT scanning, the cuttingsmay be saved for further analysis or may be suitably discarded. Anexample of a suitable CT scanner for making images usable with methodsaccording to the invention is sold under model designation MicroXCTSeries 3D tomographic x-ray transmission microscope by Xradia, Inc.,5052 Commercial Circle, Concord, Calif. 94520.

In some examples, an analysis of the cuttings from the CT scan imagesmay provide, substantially in real time during the drilling of thewellbore, an estimate of certain properties of the subsurface formationsbeing drilled, for example fluid mobility of one or more constituentfluids in the pore spaces of the rock formations 11. In the presentexample, images generated by the CT scanner 30 may be transferred to acomputer 32 having program instructions for carrying out image analysisand subsequent formation property modeling as described below.

It should also be understood that drill cuttings are only one type ofrock sample that may be analyzed according to the invention. In otherexamples, the drill bit 16 may be an annular type configured to drillwhole cores of the rock formations 11. In other examples, percussionsidewall core samples may be obtained during drilling or when the drillstring 14 is withdrawn from the wellbore 12 such as for “wireline” wellevaluation techniques. Accordingly, the scope of the invention is notlimited to analysis of drill cuttings. As explained above, the inventionis also not limited to use with rock samples obtained from a wellboredrilled through subsurface rock formations.

CT scan imaging of a porous material sample (e.g., a sample of rockformation) is used in the invention to produce a numerical object thatrepresents the material sample digitally in the computer 32 forsubsequent numerical simulations of various physical processes, such asviscous fluid flow (for permeability estimation); stress loading (forthe effective elastic moduli); electrical current flow (forresistivity); and pore size distribution for nuclear magnetic resonancerelaxation time properties, including distribution of relaxation time.In some examples, such analysis can be performed while drillingoperations are underway, substantially in real time.

The CT scan image produced by the CT scanner 30 may be a 3D numericalobject consisting of a plurality of 2D sections of the imaged sample.Each 2D section consists of a grid of values each corresponding to asmall region of space defined within the plane of the grid. Each suchsmall region of space is referred to as a “pixel” and has assignedthereto a number representing the image darkness (or for example thedensity of the material) determined by the CT scan procedure. The valueascribed to each pixel of the 2D sections is typically an integer thatmay vary between zero and 255 where 0 is, e.g., pure white, and 255 ispure black. Such integer is typically referred to as a “gray scale”value. 0 to 255 is associated with eight digital bits in a digital wordrepresenting the gray scale value in each pixel. Other gray scale rangesmay be associated with longer or shorter digital words in otherimplementations, and the range of 0 to 255 is not intended to limit thescope of the invention. For the purpose of simulating a physical processusing such a numerical object (the gray scale), however, the numericalobject is preferably processed so that all the pixels allocated to thevoid space in the rock formation (pore space) are represented by acommon numerical value, e.g., by only 255s, and all the pixelsassociated with the rock matrix (or rock grains) are represented by adifferent numerical value, for example, zeroes. The foregoing process iscalled image segmentation. Subsequently, the resulting numerical objectcan be normalized so that the pore spaces are represented by, forexample, ones and the rock grains are represented by zeroes. Theforegoing may be described as converting the image into a binary index.In other examples, the image may be converted into an index having anyselected number, n, of indices. It has been determined that sufficientlyaccurate modeling of some rock petrophysical parameters or properties,e.g. permeability, may be obtained using a binary index, in which onevalue represents pore space and another single value represents rockgrains.

A technique known in the art for segmenting a gray-scale object iscalled “thresholding”, where all pixels having a gray scale value belowa selected threshold value (e.g., a gray scale value of 150 on a scaleof 0 to 255) are identified as grains, while all other pixels areidentified as pore space. The foregoing approach is often notsatisfactory, however, because, due to numerical clutter in anunprocessed CT scan image, some pixels physically located inside a grainmay have the gray level of the pore space and vice versa. In theinvention, a type of image segmentation known as “region growing” can beused. Region growing may be described as follows. Consider a 2D sectionof a CT scan image made of a porous rock formation such as sandstone,which has primarily quartz rock grains. A substantial number of “seeds”(each seed consists of one or more pixels having a similar pixel grayscale level, e.g., 250±5) is placed within the image. All pixels withina seed are assigned the same gray scale level which may be an average(e.g., arithmetic) of the gray levels of all the pixels within the seed.The seeds in the image frame do not overlap spatially. Next, two or moreadjacent seeds are merged and are identified as a “region” if the grayscale levels of the adjacent seeds have gray scale values within aselected difference threshold of each other. Each identified region isassigned a uniform (fixed) gray level, which can be a weighted averageof the gray scale values of all the seeds that have been merged into theidentified region. The foregoing process continues for all regions thusformed in the image frame. As a result, the unprocessed CT image istransformed into internally uniform regions plus unclassified pixelsthat were not assigned to any of the identified regions (because suchpixels included gray scale values outside the allocation thresholdcriteria). Each of such unclassified pixels can be assigned to anadjacent region with the closest gray scale level. If the resultingnumber of regions is greater than two, however, the foregoing methodsimply fails to allocate the CT image correctly into grains and pores.

To address the foregoing problem with extending (“growing”) seeds intoregions, in the invention, instead of using seeds having different grayscale values, only two classes of seeds are used: all pixels having agray scale value below a selected initial limit for the gray scale levelof rock grains (e.g., 60) are classified as rock grains; and all pixelsin which the gray scale level is larger than a selected initial limitfor pore spaces (e.g., 130) are classified as pore space. One simple wayof specifying these initial limits is by selecting the gray scale levelscorresponding to the peaks of a gray level histogram. In many subsurfaceformations, such a histogram will be bimodal, wherein one mode valuewill correspond to the gray scale level of pores, and another mode valuewill correspond to the gray scale level of rock grains.

The next element in image classification according to the invention isto grow each of the two initially formed seeds by allocating to suchseeds all adjacent pixels having gray scale levels within a selectedtolerance, e.g., 130−5 for pore spaces and 60+5 for rock grains. Theforegoing process can continue by incrementally increasing the grayscale lower limit for rock grains and incrementally reducing the grayscale upper limit for pore spaces until the two limits meet. The resultis that all pixels will be allocated to either pore space or to rockgrains, thus providing a fully segmented image.

A possible advantage of the foregoing procedure is that instead offorming multiple regions, the foregoing technique grows only twodistinctive regions from start to end, thus avoiding the situation wheremultiple distinctive regions appear and then have to be reclassifiedinto either pores or grains. If the resulting segmented image appearsnoisy (cluttered), it can be smoothed by any of conventional filters.

A schematic outline of the foregoing procedure follows. First is topreprocess the original image using the median or 2D Gaussian kernelfilter. The size of the filter is provided by the user and should dependon, among other factors, the quality of the image (level of noise). Itshould be noted that the image segmenting procedure that follows hasbeen demonstrated to be sufficiently noise resistant as to make thepreprocessing frequently unnecessary.

Next, two user-selected thresholds, t₁ and t₂, are selected to determineinitial regions for pore space and rock grains, respectively. Theinitial thresholds may be selected, for example, by analysis of ahistogram of the gray scale values in the CT image. For every pixelp_(i) having a gray scale level represented by B(p_(i)):

if B(p_(i))>t_(i) then p_(i) is identified as pore space; and

if B(p_(i))<t₂ then p_(i) is identified as rock grain.

If there are two or more contiguous pixels in any subset of the imageframe that are classified according to the threshold procedure above,such contiguous pixels may be referred to as “clusters.” All of thepixels allocated as explained above then become the image seeds fromwhich region growing proceeds.

Finally, for each pixel classified as a pore, its eight neighbors(spatially contiguous pixels) in the 2D image plane are interrogated. Ifany of the interrogated neighbor pixels is not already identified aspore or rock grain, and the gray scale level of such pixel is within apreselected tolerance level of (or initially selected different between)the gray scale level assigned to the “pore” seed (as in Step 2 above),the interrogated neighbor pixel is then classified as a pore and isallocated to the “pore” cluster.

The foregoing contiguous pixel interrogation is also performed forpixels classified as rock grain. Contiguous, previously unallocatedpixels having gray scale level within a preselected tolerance of thegray scale level of the rock grain seed are allocated to the rock graincluster.

The foregoing cluster allocation and region growing process continuesfor both pore space and rock grain until all the pixels in the 2D imageframe are interrogated. If any of the pixels is not classified as porespace or rock grain, the foregoing tolerance value for each of the porespace and the rock grain may be increased by a selected increment (forexample five gray scale numbers), and the contiguous pixel interrogationand classification may be repeated. The foregoing tolerance increase andrepeated adjacent pixel interrogation may be repeated until all orsubstantially all the pixels in the 2D image frame are allocated toeither rock grain or pore space.

The foregoing region growing procedure is then repeated for each 2Dimage frame in the 3D CT scan image. The result is a three dimensionalcharacterization of the pore structure of the rock samples on which CTimaging has been performed.

An example implementation of the above process for image segmentation isshown in a flow chart in FIG. 2. At 40, a 2D image frame of a CT scanimage is selected. The image frame may be subjected to histogramanalysis, at 42 to determine possible mode values of gray scale for porespaces and for rock grains. At 44, the possible modes of the histogrammay be used to set initial values for the image segmentation thresholdst₁ and t₂. At 46, using the initial segmentation thresholds, all pixelsin the image frame are interrogated may be are allocated to pore spaceor to rock grains, depending on whether the gray scale value in eachpixel exceeds the respective segmentation threshold. The allocatedpixels are then segmented into seeds where two or more contiguous pixelsare allocated to either pore space or rock grain. At 48, pixels adjacentto the each of the seeds are interrogated. Previously unallocated pixelshaving a gray scale value falling within an initially selected thresholddifference (or tolerance) of the adjacent cluster pixel gray scale valueare allocated to the seed at 50. At 54, the image frame is interrogatedto determine if all or substantially all the image frame pixels havebeen allocated to either pore space or rock grain. At 54, the number ofallocated pixels is counted and at 60 if all or substantially all thepixels in the image frame have been allocated, a new 2D image frame canbe selected, at 58, and the above process repeated. Typically the next2D image frame will be adjacent to the most recently analyzed 2D imageframe. The above process can be repeated until all available 2D imageframes have been analyzed. If all pixels in the image frame have notbeen allocated, at 52, the tolerance or difference threshold values usedat 50 may be increased and the interrogation of pixels adjacent to theexisting seeds can be repeated, at 48, and the remainder of the processcan be repeated.

The result of the foregoing procedure is a segmented 3D image of therock sample including image elements for rock grain and for pore space.Such image can be stored or displayed in a computer and can be used asinput to one or more rock property characterization models.

The foregoing implementation of image segmentation may proveadvantageous when implementing analysis techniques according to theinvention, which require a segmented image as input for furtheranalysis. Such techniques may be explained as follows. In many cases, arelatively large range for each input parameter (e.g., porosity) can beobtained from a single sample of rock formation, for example drillcuttings (FIG. 1), sidewall cores, outcrop quarry or whole drill cores.Referring to FIG. 3, the sample of rock formation, obtained at 60,should be imaged, such as by CT-scanning, to obtain a high-resolution 3Dimage of the pore space and rock grains (matrix). The foregoing is shownat 62. Next, at 64, the image should be segmented to allocate imageportions to the rock matrix and the pore space. Image segmentation maybe performed as explained above with reference to FIG. 2.

By counting the pixels in the segmented image that are allocated to thepore space and dividing their number by the total number of pixels inthe image, an estimate of the porosity φ may be obtained, at 66. Next,at 68, a numerical simulation of a physical experiment may be conductedon the sample image. Such numerical experiments and methods to conductthese experiments may include but are not limited to: (a) single-phasefluid flow using the Lattice-Boltzmann numerical method (LBM) to obtainthe absolute permeability; (b) elastic deformation using thefinite-element method (FEM) to obtain the imaged sample's elastic moduliand elastic-wave velocity; and (c) electrical current flow using FEM toobtain the imaged sample's electrical resistivity and formation factor.The foregoing procedure will produce a single data point, such asporosity/velocity, porosity/permeability, and/or porosity/formationfactor, which is not sufficient to provide a rock physics transform forany of such data pairs.

To obtain a sufficient number of data points to establish relationshipsbetween porosity and other petrophysical parameters, the originalsegmented image volume may be subdivided into a selected number ofsub-volumes, as shown at 70. Subdividing the image can be performed bydividing the original volume into a number of evenly spaced volumes orby randomly selecting a sufficient number of sub-volumes. One example isto divide a cubic image volume into sub-cubes. Examples of sub-cubesinclude dividing the original image volume into eight, twenty seven,sixty four or one hundred twenty five cubic sub-volumes. A value ofporosity may be determined for each sub-volume, as shown at 72.

Because natural rock formations are heterogeneous at essentially allscales, it is often observed that even for a relatively homogeneousformation sample, such as Berea sandstone, the porosity and/or otherattributes of sub-volumes typically includes a relatively large range ofeach petrophysical parameter of interest, and such range is typicallysufficient to obtain a meaningful transform for each such parameter.

After subdividing the image volume and determining the porosity of eachsub-volume, numerical simulation of any one or more petrophysicalparameters of interest can be performed on each of the sub-volumes (justas performed on the entire image volume), at 72. The results may beplotted or otherwise allocated in the computed outputs (e.g., absolutepermeability k) with respect to the input parameters (e.g., porosity φ)at 76. The resulting data points often are sufficient in number toestablish a relationship between them (i.e., a transform).

Next, the data points above are used, at 78, for the one or more modeledpetrophysical parameters, to obtain a relationship such as a best-fitanalytical expression between porosity and the petrophysicalparameter(s). In a first example, the porosity may be related topermeability. The foregoing relationship determination may be repeatedwith respect to formation resistivity factor, at 80. The foregoingrelationship determination may also be repeated for bulk, elastic and/orshear moduli, at 82.

In the present example, the Lattice-Boltzmann method can be used tonumerically solve Navier-Stokes equations for flow simulation forpermeability modeling. Such solution may be used to calculatepermeability of simulated 3D volumes. The Lattice-Boltzmann method is arobust tool for flow simulation, particularly in media with complex poregeometry. See, for example. Ladd, Numerical Simulations of ParticulateSuspensions via a discretized Boltzmann Equation, Part 1: TheoreticalFoundation, J. Fluid Mech., v 271, 1994, pp. 285-309; Gunstensen et al.,“Lattice Boltzmann Model of Immiscible Fluids, Phys. Rev. A., v. 43, no.8, Apr. 15, 1991, pp. 4320-4327; Olsen et al., Two-fluid Flow inSedimentary Rock: Simulation, Transport and Complexity, J. FluidMechanics, Vol. 341, 1997, pp. 343-370; and Gustensen et al.,Lattice-Boltzmann Studies of Immiscible Two-Phase Flow Through PorousMedia,” J. of Geophysical Research, V. 98, No. B4, Apr. 10, 1993, pp.6431-6441).

The Lattice-Boltzmann method simulates fluid motion as collisions ofimaginary particles, which are much larger than actual fluid molecules,but wherein such particles show almost the same behavior at amacroscopic scale. The algorithm used in the Lattice-Boltzmann methodrepeats collisions of these imaginary particles until steady state isreached, and provides a distribution of local mass flux. In accordancewith the present invention, the Lattice-Boltzmann method is appliedsuccessfully for many pore structures, including cylindrical tubes,random densely packed spheres, and 3D rock samples digitized by CTscanning as explained above. See, for example, U.S. Pat. No. 6,516,080issued to Nur.

It is also possible to estimate capillary pressure related flowcharacteristics from the pore structure determined using the 3D imagesprocessed as explained above. See, for example, U.S. Pat. No. 7,277,795issued to Boitnott. Any or all of the foregoing estimated petrophysicalproperties and parameters may be stored and/or displayed in the computer(32 in FIG. 1).

While the invention has been described with respect to a limited numberof embodiments, those skilled in the art, having benefit of thisdisclosure, will appreciate that other embodiments can be devised whichdo not depart from the scope of the invention as disclosed herein.Accordingly, the scope of the invention should be limited only by theattached claims.

1. A method for estimating a relationship between physical properties ofa porous material from a sample thereof, comprising: making a threedimensional tomographic image of the sample of the material; segmentingthe image into pixels each representing pore space or mineral grain;subdividing the image into sub-volumes; estimating a porosity for eachsub-volume from the image thereof; modeling at least one petrophysicalparameter from the image of each sub-volume; determining a relationshipbetween the porosity and the modeled petrophysical parameter using theestimated porosity and the modeled petrophysical parameter from aplurality of the sub volumes; and at least one of storing and displayingthe relationship and the modeled petrophysical parameter.
 2. The methodof claim 1 wherein the tomographic image comprises a value of gray scaleallocated to each of a plurality of pixels in the image.
 3. The methodof claim 2 wherein the segmenting comprises: (a) determining an initialgray scale threshold for each of pore space and rock grain; (b)allocating each pixel in the image to pore space or rock grain for eachpixel meeting threshold criteria for each of the pore space and rockgrain thresholds, respectively; (c) interrogating pixels adjacent toeach seed; (d) allocating the interrogated adjacent pixels previouslynot allocated to a seed to the pore space seed or the rock grain seedbased on threshold criteria; and (e) repeating (c) and (d) untilsubstantially all pixels in the image are allocated to the rock grain orthe pore space.
 4. The method of claim 3 wherein the determining initialgray scale thresholds comprises histogram analysis of the tomographicimage.
 5. The method of claim 3 wherein the allocating interrogatedadjacent pixels comprises determining a difference between a gray scalevalue and a gray scale value of the adjacent pixel in the seed, andallocating the interrogated pixel to the seed if the difference fallsbelow a selected threshold.
 6. The method of claim 5 further comprising:determining whether unallocated pixels exist in the image; increasingthe selected difference threshold; and repeating the interrogatingadjacent and allocating the interrogate pixels having gray scale valueswherein the difference is below the increased difference threshold. 7.The method of claim 1 wherein the material comprises a rock formation.8. The method of claim 7 wherein the at least one petrophysicalparameter comprises permeability.
 9. The method of claim 8 whereinpermeability is estimated using the Lattice-Boltzmann approximation. 10.The method of claim 7 wherein the at least one petrophysical parametercomprises electrical resistivity formation factor.
 11. The method ofclaim 7 wherein the at least one petrophysical parameter comprises atleast one of elastic modulus, bulk modulus and shear modulus.
 12. Themethod of claim 1 wherein the relationship is determined by statisticalbest fit.